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Градиентный бустинг. Реализация с нуля на Python и разбор особенностей его модификаций (XGBoost, CatBoost, LightGBM)

На сегодняшний день градиентный бустинг (gradient boosting machine) является одним из основных production-решений при работе с табличными, неоднородными данными, поскольку обладает высокой производительностью и точностью, а если быть точнее, то его модификации, речь о которых пойдёт чуть позже.

В данной статье представлена не только реализация градиентного бустинга GBM с нуля на Python, но а также довольно подробно описаны ключевые особенности его наиболее популярных модификаций.

Ноутбук с данными алгоритмами можно загрузить на Kaggle (eng) и GitHub (rus).

Содержание

Gradient Boosting Machine

Gradient Boosting Machine

Как и в случае с Adaboost, градиентный бустинг добавляет базовые модели в ансамбль последовательно, однако вместо обучения моделей с учётом весов на основе ошибок предшественников, в данном случае модели обучаются на остаточных ошибках (residual errors), допущенных предыдущими моделями.

Принцип работы градиентного бустинга для регрессии

Алгоритм строится следующим образом:

1) первоначальному прогнозу присваивается среднее значение y_train для всех образцов;
2) рассчитываются остатки модели на основе антиградиента функции потерь;
3) регрессионное дерево обучается на X_train и остатках, далее делается прогноз на X_train;
4) полученный прогноз добавляется к первоначальному и шаги 2-4 повторяются для каждого дерева;
5) после обучения всех моделей снова создаётся первоначальный прогноз из шага 1;
6) далее делаются прогнозы для X_test на обученных деревьях и добавляются к первоначальному;
7) полученная сумма и будет конечным прогнозом.

Формулы для расчётов

Функция потерь: $L(y_i, F(x_i)) = \frac{1}{2} (y_i - F(x_i))^2$

Остатки: $r_{ik} = -\left[\frac{\partial L(y_i, F(x_i))}{\partial F(x_i)}\right]_ {F(x) = F_{m-1}(x)} = y_{i} - F_{m-1}(x_i)$

Принцип работы градиентного бустинга для классификации

В случае классификации дела обстоят немного сложнее: к каждому уникальному классу необходимо применить one-hot encoding и перевести в вероятности с помощью softmax-функции, а к прогнозам деревьев добавить коэффициент $\gamma$ , который регулирует степень вклада каждого нового дерева в общую модель для снижения переобучения. Также качество прогнозов градиентного бустинга можно существенно улучшить, применив концепцию K-class LogitBoost: для каждого дерева рассчитываются веса, а только потом на их основе остатки.

Алгоритм строится следующим образом:

1) для y_train применяется one-hot encoding и первоначальному прогнозу приcваиваются значения 0 для каждого класса;
2) прогнозы преобразуются в вероятности с помощью softmax-функции;
3) рассчитываются остатки модели на основе антиградиента функции потерь и вероятностей (а также веса в случае LogitBoost);
4) регрессионное дерево обучается на X_train и остатках (а в случае LogitBoost ещё присваиваются веса), далее делается прогноз на X_train;
5) для каждого листа в дереве рассчитываются коэффициенты $\gamma$ на основе остатков, взятых по позициям наблюдений, попавших в определённый в листовой узел;
6) полученные прогнозы для каждого класса и сумма коэффициентов $\gamma$ добавляются к первоначальным;
7) шаги 2-6 повторяются для каждого дерева в каждом классе;
8) после обучения всех моделей создаётся первоначальный прогноз из шага 1;
9) далее делаются прогнозы для X_test на обученных деревьях по каждому классу и добавляются к первоначальным;
10) классы с максимальной суммой и будут конечным прогнозом.

Формулы для расчётов

Функция потерь:

$L(\{y_k, F_k(x)\}_{1}^{K}) = -\sum\limits_{k=1}^{K} y_k \ log \ p_k(x)$

Вероятность каждого класса:

$p_k(x) = softmax(F_k(x)) = \frac{e^{F_k(x)}}{\sum\limits_{l=1}^{K} e^{F_l(x)}}$

Остатки:

$r_{ik} = -\left[\frac{\partial L(\{y_{il}, F_l(x_i)\}_{l=1}^{K})}{\partial F_k(x_i)}\right]_ {\{F_l(x) = F_{l,m-1}(x)\}_{1}^{K}} = y_{ik} - p_{k,m-1}(x_i)$

Коэффициент $\gamma$ :

$\gamma_{jkm} = \frac{K-1}{K} \frac{\sum_{x_i \in R_{jkm}} r_{ik}} {\sum_{x_i \in R_{jkm}} |r_{ik}| (1 - |r_{ik}|)}$

Веса для K-class LogitBoost:

Остатки для K-class LogitBoost:

$r_{ik} = \frac{K-1}{K} \frac{y_{ik} - p_k(x_i)}{w_k(x_i)}$

Импорт необходимых библиотек

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.datasets import load_diabetes
from sklearn.preprocessing import LabelEncoder
from sklearn.model_selection import train_test_split
from sklearn.metrics import accuracy_score, mean_absolute_percentage_error
from sklearn.tree import DecisionTreeRegressor
from sklearn.ensemble import GradientBoostingClassifier, GradientBoostingRegressor
from sklearn.ensemble import HistGradientBoostingClassifier, HistGradientBoostingRegressor
from mlxtend.plotting import plot_decision_regions

Реализация на Python с нуля

class GBMClassifier:
    def __init__(self, logitboost=False, learning_rate=0.1, n_estimators=100,
                 max_depth=3, random_state=0):
        self.logitboost = logitboost
        self.learning_rate = learning_rate
        self.n_estimators = n_estimators
        self.max_depth = max_depth
        self.random_state = random_state

    def _softmax(self, predictions):
        exp = np.exp(predictions)

        return exp / np.sum(exp, axis=1, keepdims=True)

    def _compute_gammas(self, residuals, leaf_indexes, eps=1e-4):
        gammas = []

        for j in np.unique(leaf_indexes):
            x_i = np.where(leaf_indexes == j)
            numerator = np.sum(residuals[x_i])
            norm_residuals_xi = np.linalg.norm(residuals[x_i]) + eps
            denominator = np.sum(norm_residuals_xi * (1 - norm_residuals_xi))
            gamma = (self.K - 1) / self.K * numerator / denominator
            gammas.append(gamma)

        return gammas

    def fit(self, X, y):
        self.K = len(np.unique(y))
        self.trees = {k: [] for k in range(self.K)}
        one_hot_y = pd.get_dummies(y).to_numpy()   # one-hot encoding
        predictions = np.zeros(one_hot_y.shape)

        for _ in range(self.n_estimators):
            probabilities = self._softmax(predictions)

            for k in range(self.K):
                if self.logitboost:   # based on K-class LogitBoost
                    numerator = (one_hot_y.T[k] - probabilities.T[k])
                    denominator = probabilities.T[k] * (1 - probabilities.T[k])
                    residuals = (self.K - 1) / self.K * numerator / denominator
                    weights = denominator
                else:
                    residuals = one_hot_y.T[k] - probabilities.T[k]
                    weights = None

                tree = DecisionTreeRegressor(criterion='friedman_mse', max_depth=self.max_depth,
                                             random_state=self.random_state)
                tree.fit(X, residuals, sample_weight=weights)
                self.trees[k].append(tree)

                leaf_indexes = tree.apply(X)
                gammas = [] if self.logitboost else self._compute_gammas(residuals, leaf_indexes)
                predictions.T[k] += self.learning_rate * tree.predict(X) + np.sum(gammas)

    def predict(self, samples):
        predictions = np.zeros((len(samples), self.K))

        for i in range(self.n_estimators):
            for k in range(self.K):
                predictions.T[k] += self.learning_rate * self.trees[k][i].predict(samples)

        return np.argmax(predictions, axis=1)

class GBMRegressor:
    def __init__(self, learning_rate=0.1, n_estimators=100, max_depth=3, random_state=0):
        self.learning_rate = learning_rate
        self.n_estimators = n_estimators
        self.max_depth = max_depth
        self.random_state = random_state
        self.trees = []

    def fit(self, X, y):
        self.initial_leaf = y.mean()
        predictions = np.zeros(len(y)) + self.initial_leaf

        for _ in range(self.n_estimators):
            residuals = y - predictions
            tree = DecisionTreeRegressor(criterion='friedman_mse', max_depth=self.max_depth,
                                         random_state=self.random_state)
            tree.fit(X, residuals)
            predictions += self.learning_rate * tree.predict(X)
            self.trees.append(tree)

    def predict(self, samples):
        predictions = np.zeros(len(samples)) + self.initial_leaf

        for i in range(self.n_estimators):
            predictions += self.learning_rate * self.trees[i].predict(samples)

        return predictions

Код для отрисовки решающих границ

def decision_boundary_plot(X, y, X_train, y_train, clf, feature_indexes, title=None):
    feature1_name, feature2_name = X.columns[feature_indexes]
    X_feature_columns = X.values[:, feature_indexes]
    X_train_feature_columns = X_train.values[:, feature_indexes]
    clf.fit(X_train_feature_columns, y_train.values)

    plot_decision_regions(X=X_feature_columns, y=y.values, clf=clf)
    plt.xlabel(feature1_name)
    plt.ylabel(feature2_name)
    plt.title(title)

Загрузка датасетов

Для обучения моделей будет использован Glass Classification датасет, где необходимо верно определить тип стекла по его признакам. В случае регрессии используется Diabetes датасет из scikit-learn.

Данные ниже будут использоваться в примерах для всех модификаций градиентного бустинга.

df_path = "/content/drive/MyDrive/glass.csv"
glass_df = pd.read_csv(df_path)
X1, y1 = glass_df.iloc[:, :-1], glass_df.iloc[:, -1]
y1 = pd.Series(LabelEncoder().fit_transform(y1))
X1_train, X1_test, y1_train, y1_test = train_test_split(X1, y1, random_state=0)
print(glass_df)


          RI     Na    Mg    Al     Si     K    Ca    Ba   Fe  Type
0    1.52101  13.64  4.49  1.10  71.78  0.06  8.75  0.00  0.0     1
1    1.51761  13.89  3.60  1.36  72.73  0.48  7.83  0.00  0.0     1
2    1.51618  13.53  3.55  1.54  72.99  0.39  7.78  0.00  0.0     1
3    1.51766  13.21  3.69  1.29  72.61  0.57  8.22  0.00  0.0     1
4    1.51742  13.27  3.62  1.24  73.08  0.55  8.07  0.00  0.0     1
..       ...    ...   ...   ...    ...   ...   ...   ...  ...   ...
209  1.51623  14.14  0.00  2.88  72.61  0.08  9.18  1.06  0.0     7
210  1.51685  14.92  0.00  1.99  73.06  0.00  8.40  1.59  0.0     7
211  1.52065  14.36  0.00  2.02  73.42  0.00  8.44  1.64  0.0     7
212  1.51651  14.38  0.00  1.94  73.61  0.00  8.48  1.57  0.0     7
213  1.51711  14.23  0.00  2.08  73.36  0.00  8.62  1.67  0.0     7

[214 rows x 10 columns]

X2, y2 = load_diabetes(return_X_y=True, as_frame=True)
X2_train, X2_test, y2_train, y2_test = train_test_split(X2, y2, random_state=0)
print(X2, y2, sep='\n')


          age       sex       bmi        bp        s1        s2        s3  \
0    0.038076  0.050680  0.061696  0.021872 -0.044223 -0.034821 -0.043401   
1   -0.001882 -0.044642 -0.051474 -0.026328 -0.008449 -0.019163  0.074412   
2    0.085299  0.050680  0.044451 -0.005670 -0.045599 -0.034194 -0.032356   
3   -0.089063 -0.044642 -0.011595 -0.036656  0.012191  0.024991 -0.036038   
4    0.005383 -0.044642 -0.036385  0.021872  0.003935  0.015596  0.008142   
..        ...       ...       ...       ...       ...       ...       ...   
437  0.041708  0.050680  0.019662  0.059744 -0.005697 -0.002566 -0.028674   
438 -0.005515  0.050680 -0.015906 -0.067642  0.049341  0.079165 -0.028674   
439  0.041708  0.050680 -0.015906  0.017293 -0.037344 -0.013840 -0.024993   
440 -0.045472 -0.044642  0.039062  0.001215  0.016318  0.015283 -0.028674   
441 -0.045472 -0.044642 -0.073030 -0.081413  0.083740  0.027809  0.173816   

           s4        s5        s6  
0   -0.002592  0.019907 -0.017646  
1   -0.039493 -0.068332 -0.092204  
2   -0.002592  0.002861 -0.025930  
3    0.034309  0.022688 -0.009362  
4   -0.002592 -0.031988 -0.046641  
..        ...       ...       ...  
437 -0.002592  0.031193  0.007207  
438  0.034309 -0.018114  0.044485  
439 -0.011080 -0.046883  0.015491  
440  0.026560  0.044529 -0.025930  
441 -0.039493 -0.004222  0.003064  

[442 rows x 10 columns]
0      151.0
1       75.0
2      141.0
3      206.0
4      135.0
       ...  
437    178.0
438    104.0
439    132.0
440    220.0
441     57.0
Name: target, Length: 442, dtype: float64

Обучение моделей и оценка полученных результатов

В случае классификации ручная реализация немного отличается в плане точности от scikit-learn — это связано с тем, что в последней применяется более усложнённый подход для прогнозирования и расчёта остатков, но суть остаётся прежней, и в большинстве случаев работает примерно также как и ручная реализация, особенно в сравнении с LogitBoost модификацией.

В случае регрессии, ошибка в ручной реализации даже немного ниже, чем в scikit-learn — различия незначительны.

Также стоит отметить, что в scikit-learn есть особая версия градиентного бустинга — HistGradientBoosting на основе LightGBM, предназначенная для работы с большими наборами данных, которая отрабатывает гораздо быстрее и чаще лучше, чем классический градиентный бустинг.

Полученные результаты приведены ниже.

GBMClassifier

gbc = GBMClassifier(random_state=0)
gbc.fit(X1_train, y1_train)
gbc_pred_res = gbc.predict(X1_test)
gbc_accuracy = accuracy_score(gbc_pred_res, y1_test)
print(f'gbc accuracy: {gbc_accuracy}')
print(gbc_pred_res)


gbc accuracy: 0.6111111111111112
[5 0 1 4 1 1 0 1 1 1 1 0 1 1 1 5 0 1 1 1 5 0 5 5 1 0 5 0 1 0 0 4 1 0 0 0 0
 0 0 5 1 4 1 0 0 1 0 1 0 1 0 3 5 0]

GBMClassifier с концепцией LogitBoost

logit_bc = GBMClassifier(logitboost=True, random_state=0)
logit_bc.fit(X1_train, y1_train)
logit_bc_pred_res = logit_bc.predict(X1_test)
logit_bc_accuracy = accuracy_score(logit_bc_pred_res, y1_test)
print(f'logit_gbc accuracy: {logit_bc_accuracy}')
print(logit_bc_pred_res)


logit_gbc accuracy: 0.6666666666666666
[5 0 1 4 1 1 0 1 1 1 1 0 0 1 1 5 2 1 1 0 3 0 5 1 2 0 5 0 0 1 0 0 1 0 0 0 0
 2 0 5 1 4 1 0 1 1 0 1 0 1 0 1 5 0]

GradientBoostingClassifier (scikit-learn)

sk_gbc = GradientBoostingClassifier(random_state=0)
sk_gbc.fit(X1_train, y1_train)
sk_gbc_pred_res = sk_gbc.predict(X1_test)
sk_gbc_accuracy = accuracy_score(sk_gbc_pred_res, y1_test)
print(f'sk_gbc accuracy: {sk_gbc_accuracy}')
print(sk_gbc_pred_res)

feature_indexes = [1, 3]
title1 = 'GradientBoostingClassifier surface'
decision_boundary_plot(X1, y1, X1_train, y1_train, sk_gbc, feature_indexes, title1)


sk_gbc accuracy: 0.6666666666666666
[5 0 1 4 1 1 0 1 1 1 1 0 1 1 1 5 2 1 2 0 1 0 5 5 0 0 5 0 0 1 0 0 1 0 1 0 0
 0 0 5 1 1 1 0 1 1 0 1 0 1 0 1 5 0]

HistGradientBoostingClassifier (scikit-learn)

sk_hist_gbc = HistGradientBoostingClassifier(random_state=0)
sk_hist_gbc.fit(X1_train, y1_train)
sk_hist_gbc_pred_res = sk_hist_gbc.predict(X1_test)
sk_hist_gbc_accuracy = accuracy_score(sk_hist_gbc_pred_res, y1_test)
print(f'sk_hist_gbc accuracy: {sk_hist_gbc_accuracy}')
print(sk_hist_gbc_pred_res)

feature_indexes = [1, 3]
title2 = 'HistGradientBoostingClassifier surface'
decision_boundary_plot(X1, y1, X1_train, y1_train, sk_hist_gbc, feature_indexes, title2)


sk_hist_gbc accuracy: 0.7407407407407407
[5 0 1 4 1 1 0 1 1 1 0 1 0 1 1 5 2 1 0 2 3 0 5 5 0 0 5 0 1 1 0 0 1 0 0 0 0
 2 0 5 1 4 1 0 0 1 0 1 0 1 0 1 5 0]

GBMRegressor

gbr = GBMRegressor(random_state=0)
gbr.fit(X2_train, y2_train)
gbr_pred_res = gbr.predict(X2_test)
mape = mean_absolute_percentage_error(gbr_pred_res, y2_test)
print(f'gbr mape: {mape}')
print(gbr_pred_res)


gbr mape: 0.3422881238725283
[255.26058697 236.87383554 189.22369747 111.4291292  195.31958776
 249.2293251   97.46956406 205.91110801 133.47952304 240.88679883
 187.19018483 171.09963906 132.63933991  86.28964353 286.56077589
 105.78347251 161.94686043  72.58069452 105.92470096 236.52718059
 178.31146418 127.93036968 164.16563703 147.68681371 229.72747212
 194.16656023 142.44300193  68.79177464 237.21638657 163.36730299
 228.0338107   85.82843049 122.42781509 153.99332065 133.44848601
 159.00754731 151.98981608 147.1238119   88.56372077 200.84785209
 111.32129613 157.60108633 137.94395538 173.39402287 176.29284757
  62.85019316 109.646804   124.14982054  83.45697583 266.40618239
 141.2307366   42.05033124 163.57898214 149.92213387 230.37326368
 226.98842404 176.95426024 128.73302071  94.7641018  180.13853236
 242.31821245 147.52903035 130.45931832  97.32617479 246.70709949
 155.26338705  88.36785447 243.39330218 215.93042351  57.89243255
  87.76236479 138.21222825 101.23449416 140.80521012 128.63575744
 180.15062966 126.14000544 220.28212417 272.05642136 203.4384734
 135.96206295 180.87259854  55.62066017 241.84218009 137.19320157
  85.57441147 154.16763601 198.15465153 107.34659467 158.72672412
 102.17554103 114.4602227   70.44016611 165.66282053 108.02991765
  87.88855231 218.47152891 250.1208086  131.93276827 164.8880371
 174.16393905 119.21512394 150.9405975   93.44225978 234.7444088
 139.46581273 248.28728947 276.8291188  121.36081625  77.12248695
 252.1021757 ]

GradientBoostingRegressor (scikit-learn)

sk_gbr = GradientBoostingRegressor(random_state=0)
sk_gbr.fit(X2_train, y2_train)
sk_gbr_pred_res = sk_gbr.predict(X2_test)
sk_mape = mean_absolute_percentage_error(sk_gbr_pred_res, y2_test)
print(f'sk_gbr mape: {sk_mape}')
print(sk_gbr_pred_res)


sk_gbr mape: 0.34315075601365186
[255.26058697 236.87383554 189.22369747 111.4291292  195.31958776
 249.2293251   97.46956406 205.91110801 133.47952304 240.88679883
 187.19018483 171.09963906 132.63933991  86.28964353 286.56077589
 105.78347251 161.94686043  72.58069452 105.92470096 236.52718059
 182.08417504 127.93036968 164.16563703 147.68681371 229.72747212
 194.16656023 142.44300193  68.79177464 237.21638657 163.36730299
 223.14955275  85.82843049 122.42781509 153.99332065 133.44848601
 159.00754731 151.98981608 147.1238119   88.56372077 200.84785209
 117.71059889 157.60108633 137.94395538 173.39402287 176.29284757
  62.85019316 109.646804   124.14982054  83.45697583 266.40618239
 141.2307366   42.05033124 163.57898214 149.92213387 230.37326368
 226.98842404 176.95426024 128.73302071  94.7641018  180.13853236
 242.31821245 147.52903035 131.71442058  97.32617479 246.70709949
 155.26338705  88.36785447 243.39330218 215.93042351  54.04270734
  87.76236479 138.21222825 101.23449416 140.80521012 128.63575744
 180.15062966 126.14000544 220.28212417 272.05642136 203.4384734
 135.96206295 180.87259854  55.62066017 241.84218009 137.19320157
  85.57441147 154.16763601 198.15465153 107.34659467 158.72672412
 102.17554103 114.4602227   70.44016611 165.66282053 108.02991765
  87.88855231 218.47152891 247.93843617 131.93276827 164.8880371
 174.16393905 119.21512394 150.9405975   93.44225978 234.7444088
 139.46581273 248.28728947 286.90898175 121.36081625  77.12248695
 252.1021757 ]

HistGradientBoostingRegressor (scikit-learn)

sk_hist_gbr = HistGradientBoostingRegressor(random_state=0)
sk_hist_gbr.fit(X2_train, y2_train)
sk_hist_gbr_pred_res = sk_hist_gbr.predict(X2_test)
sk_hist_mape = mean_absolute_percentage_error(sk_hist_gbr_pred_res, y2_test)
print(f'sk_hist_gbr mape: {sk_hist_mape}')
print(sk_hist_gbr_pred_res)


sk_hist_gbr mape: 0.3625770694795175
[264.59007545 243.59190524 180.62442773 108.53165744 195.17695513
 258.1710931  112.67894454 239.18006052 144.50027011 244.02461776
 180.19087697 171.2628973  142.6966707   94.26236013 301.44493762
  90.25622203 154.82566511  75.43291346  96.36987316 230.14565012
 196.50928346 134.41863815 167.28719774 118.51218378 209.75431111
 176.46248808 113.40188129  68.47515467 248.57638999 116.15751993
 195.54397416  59.94733366 153.03012884 165.84819847 135.95402508
 133.54279177 190.26058998 153.08963938  86.65214731 185.07312005
 110.00197459 142.99434304 111.32566034 205.63519803 163.76035901
  68.26003406 118.15074567 101.01137587  57.54356002 247.29839792
 140.32714196  41.87878548 149.89720974 189.08931404 251.48016218
 175.76119949 160.38118455 110.76716142 107.00114228 154.75409577
 256.3194536  148.8935956  119.32752592  78.65118163 252.18315433
 122.833807    92.10033912 219.56310647 199.35553751  87.27114699
  78.83529151 114.14943246  86.07433503 110.44443265 126.1152782
 193.58960355 125.06171193 207.62012728 297.41734601 206.57210109
 121.92170504 194.37322797  69.66978916 260.97214455  99.77944002
 104.34927662 131.51094535 200.0857846  114.70076625 144.92225838
 107.08233694  84.70304021  65.20782844 156.77203859  93.87016441
 117.10361355 213.79169982 258.68468943 129.45477593 140.06443861
 176.72311312 107.95798342 172.45179867 115.75603211 240.83612061
 102.2339954  249.6969811  293.80624063 110.47825419  73.86099524
 238.42985813]

Преимущества и недостатки градиентного бустинга (GBM)

Преимущества:

высокая точность;
хорошо работает с неоднородными данными;
лежит в основе других модификаций градиентного бустинга.

Недостатки:

низкая скорость обучения при работе с большими датасетами;
отсутствие возможности работы с категориальными признаками;
возможность переобучения из-за плохой предобработки данных.

Стоит отметить, что вышеперечисленные недостатки были устранены частично либо полностью в модификациях градиентного бустинга, про которые сейчас и пойдёт речь.

Дополнительные источники

Статья «Greedy Function Approximation: A Gradient Boosting Machine», Jerome H. Friedman.

Документация:

Лекции:

Пошаговое построение Gradient Boosting:

Особенности XGBoost

XGBoost (Extreme Gradient Boosting) является, пожалуй, самой популярной модификацией градиентного бустинга на сегодняшний день, с помощью которой было выиграно множество соревнований на Kaggle. Рассмотрим причины по которым так произошло.

В отличие от GBM в XGboost изначальный прогноз регрессионного дерева устанавливается f_0 (x) = 0.5 и вместо пней с глубиной 3 берутся более глубокие деревья с листьями в диапазоне от 8 до 32 как в случае регрессии, так и классификации, а к функции потерь добавляется регуляризация (L1 или L2 на выбор), которая штрафует дерево с высокой суммой нормы значений в листьях. Данная сумма представлена в виде целевой функции, расчёт значений которой аппроксимируется разложением в ряд Тейлора 2-го порядка.

Формулы для расчётов

Целевая функция:

$Obj^{\ t} = \sum\limits_{i=1}^{N} L(y_i, \hat{y_i}^{(t-1)} + f_t(x_i)) + \Omega(f_t)$

L2-регуляризация:

$\Omega(f_t) = \gamma \ T + \frac{1}{2} \lambda||w||^2$

Аппроксимация целевой функции разложением в ряд Тейлора 2-го порядка:

$Obj^{\ t} \approx \sum\limits_{i=1}^{N} [L(y_i, \hat{y_i}^{(t-1)}) + g_i f_t(x_i) + \frac{1}{2} h_i f_t^2(x_i) ] + \gamma \ T + \frac{1}{2} \lambda \sum\limits_{j=1}^{T} w_j^2 = \\ = \sum\limits_{j=1}^{T} [(\sum\limits_{i \in I_j}g_i) w_j + \frac{1}{2} (\sum\limits_{i \in I_j}h_i + \lambda) w_j^2 ] + \gamma \ T$

Где:

градиент первого порядка: ;
градиент второго порядка (гессиан): ;
параметры регуляризации: $\gamma, \lambda$ .

Приближение второго порядка удобно с вычислительной точки зрения, поскольку большинство членов одинаковы на каждой отдельной итерации и большая часть выражения может быть вычислена один раз и повторно использована как константа для всех остальных разбиений, и в таком случае единственное, что остаётся вычислить — это f_t(x_i) и $\Omega(f_t)$ .

Оптимальное значение веса j-го листового узла в дереве рассчитывается следующим образом:

$w^*_j = -\frac{\sum_{i \in I_j} g_i}{\sum_{i \in I_j} h_i + \lambda}$

Мера загрязнённости представляет из себя качество структуры дерева q и рассчитывается следующим образом:

$Obj^{\ t} = -\frac{1}{2} \sum\limits_{j=1}^{T} \frac{(\sum_{i \in I_j} g_i)^2}{\sum_{i \in I_j} h_i + \lambda} + \gamma \ T$

Чем меньше это значение, тем лучше структура дерева.

Поскольку перечисление всевозможных древовидных структур и выбор наилучшей весьма осложнён, мы можем оптимизировать данную задачу, добавляя по одному разбиению узла дерева за раз. В частности, мы итеративно добавляем ветви к дереву, используя жадный алгоритм, а информационный прирост будет выглядеть следующим образом:

$gain = \frac{1}{2} \left[\frac{(\sum_{i \in I_L} g_i)^2}{\sum_{i \in I_L} h_i + \lambda} + \frac{(\sum_{i \in I_R} g_i)^2}{\sum_{i \in I_R} h_i + \lambda} - \frac{(\sum_{i \in I} g_i)^2}{\sum_{i \in I} h_i + \lambda} \right] - \gamma$

Информационный прирост будет показывать изменение чистоты узла после его разбиения. Его максимальное значение будет соответствовать наилучшему пороговому значению (критерию разбиения) узла в дереве.

Стоит также упомянуть, что в XGBoost поддерживается выбор линейных моделей в качестве базовых, но, в целом, они работают хуже деревьев.

Дополнительные оптимизации в XGBoost

Weighted quantile sketch

В случае работы с большими наборами данных в XGBoost используется приближённый жадный алгоритм. Это означает, что вместо тестирования всех возможных пороговых значений мы тестируем только квантили. Используя распараллеливание для разделения набора данных и создания ветвей в каждом дереве независимо друг от друга чтобы несколько ядер могли работать с ними одновременно, алгоритм взвешенных квантилей (weighted quantile sketch) объединяет данные в приближённую гистограмму, разделённую на взвешенные квантили, которые помещают наблюдения с прогнозами низкой достоверности в квантили с меньшим количеством наблюдений.

Cache-aware access & out-of-core computing

Для более быстрого вычисления прогнозов градиенты и гессианы хранятся в кэш-памяти (cache-aware access), а в случае с датасетами большого размера, когда недостаточно кэш и оперативной памяти, задействуется память из жёсткого диска. Учитывая, что чтение данных на жёстком диске выполняется гораздо медленнее чем в кэш-памяти, для ускоренной обработки производится их компрессия. Данные хранятся в единицах памяти, называемых блоками, информация в которых хранится в формате сжатых столбцов, отсортированных по соответствующим значениям параллельно с использованием всех доступных ядер ЦП, а распаковка каждого блока осуществляется в независимом потоке при загрузке в память. Данная техника называется "вычисление вне ядра" (out-of-core computing). При наличии нескольких жёстких дисков, применяется параллельное сегментирование блоков для каждого жёсткого диска (sharding).

При использовании сжатых данных необходимо учитывать, что процесс сжатия и распаковки занимает дополнительное время, поэтому выбор алгоритма сжатия (с потерями и без) и его параметров может повлиять на производительность в целом, однако это всё равно значительно быстрее чтения без сжатия.

Sparsity-aware split finding

В XGboost также поддерживается работа с пропусками. Суть заключается в следующем: из исходного датасета убираются наблюдения с пропусками и представляются как отдельный датасет. После построения дерева для наблюдений без пропусков рассчитываются их остатки, далее — остатки для наблюдений с пропусками, которые присоединяются к остаткам дерева как для левого, так и для правого узла. В конце рассчитывается информационный прирост (gain) для каждого узла в дереве: выбирается порог для разбиения с максимальным приростом. Данная техника называется "поиск разбиения с учётом разряженности" (sparsity-aware split finding).

Обучение на GPU

Для ускорения процесса обучения в XGBoost реализована поддержка GPU с помощью параметров device="cuda" и tree_method="gpu_hist". Также стоит отметить, что в данном алгоритме поддерживается полностью распределённое обучение на GPU с использованием Dask, Spark и PySpark.

Пакетная реализация XGBoost и оценка полученных результатов

Благодаря вышеуказанным оптимизациям и особенностям XGBoost работает лучше классической реализации, что видно на примере классификации, однако на примере регрессии в данном случае результат оказался хуже. Если хорошо подготовить данные и поэкспериментировать с настройкой гиперпараметров, XGBoost будет работать гораздо лучше как в случае регрессии, так и классификации.

XGBClassifier

from xgboost import XGBClassifier, XGBRegressor

xgbc = XGBClassifier(seed=0)
xgbc.fit(X1_train, y1_train)
xgbc_pred_res = xgbc.predict(X1_test)
xgbc_accuracy = accuracy_score(xgbc_pred_res, y1_test)
print(f'xgbc accuracy: {xgbc_accuracy}')
print(xgbc_pred_res)

feature_indexes = [1, 3]
title2 = 'XGBClassifier surface'
decision_boundary_plot(X1, y1, X1_train, y1_train, xgbc, feature_indexes, title2)


xgbc accuracy: 0.6851851851851852
[5 0 1 4 1 1 0 1 1 1 3 0 1 1 1 5 2 1 1 2 3 0 5 5 0 0 5 0 0 1 0 0 1 0 0 0 0
 0 0 5 1 4 1 0 1 1 0 1 0 1 0 1 5 0]

XGBRegressor

xgbr = XGBRegressor(seed=0)
xgbr.fit(X2_train, y2_train)
xgbr_pred_res = xgbr.predict(X2_test)
xgbr_mape = mean_absolute_percentage_error(xgbr_pred_res, y2_test)
print(f'xgbr mape: {xgbr_mape}')
print(xgbr_pred_res)


xgbr mape: 0.3781388070785457
[281.3304   241.21165  212.13124  136.89372  144.54208  261.7871
 125.257454 236.16089  199.79964  250.18103  188.50882  126.724464
 161.2112    73.98386  302.50272   97.35103  160.85774   67.10394
  96.0406   284.72873  181.48717  117.764626 182.91393  127.76686
 229.30229  173.58592  170.03932   76.73085  240.7698   117.70699
 224.36049   63.628925 124.27327  142.07358  173.09418  139.11557
 163.61208   92.8561    95.91142  217.65016  112.03522  122.800545
 110.06953  181.34201  159.39062   61.97588  114.534805 145.20259
  84.905815 284.77567  195.29976   45.550606  92.54941  180.4689
 279.12097  241.1343   177.77785  130.43887  159.53189  158.2313
 253.25339  181.86093  120.71679   90.604706 274.5059   140.21674
  92.80314  211.71936  233.20622  110.791824  67.89235  133.62137
 123.05923  102.70884  167.2115   159.32262  131.7305   243.41493
 254.97531  210.0554   130.00981  222.21094   52.864147 251.23187
 118.602684  87.89112  121.147095 218.15724   90.9274    91.69777
 101.95428   85.79966   79.59963  155.879    118.25745   99.5975
 231.70824  189.6652   121.97507  147.51166  152.10051  102.239395
 177.74615  107.55845  271.80927  121.41044  232.97308  292.89096
 129.39447   60.501278 259.46295 ]

Преимущества и недостатки XGBoost

Преимущества:

относительно высокая скорость обучения и точность прогнозов;
возможность работы с пропусками;
гибкость в конфигурации из-за большого количества гиперпараметров.

Недостатки:

сложность в конфигурации из-за большого количества гиперпараметров;
на данный момент отсутствует полноценная (присутствует экспериментальная) возможность работы с категориальными признаками, что может сильно снизить производительность при кодировании больших объёмов данных.

Дополнительные источники

Статья «XGBoost: A Scalable Tree Boosting System», Tianqi Chen, Carlos Guestrin.

Документация.

Лекции:

один;
два.

Пошаговое построение XGBoost:

Особенности CatBoost

CatBoost (Categorical Boosting) — более эффективная реализация градиентного бустинга от компании Яндекс, особенностью которой является возможность работы с категориальными признаками без их предварительной обработки. Разберёмся подробнее как это происходит.

Ordered target (CatBoost) encoding

При кодировании категориальных признаков к бустингам применяется один из нескольких классических подходов:

target encoding;
one-hot encoding (хорошо работает лишь в том случае, когда в признаках содержится небольшое количество значений);
feature hashing в несколько корзин (исходные данные преобразуются в уникальную последовательность символов (хэш-значения), которые разбиваются на группы (корзины) и обрабатываются параллельно с последующим применением target или one-hot encoding);
mean target encoding (рассчитывается новое значение для текущего категориального признака как среднее значение целевой переменной по датасету);
leave-one-out (рассчитывается среднее значение целевой переменной по всем объектам кроме текущего);
leave-bucket-out (рассчитывается среднее значение целевой переменной на одной части данных и применяется для оставшихся других);

Однако все вышеперечисленные способы так или иначе приводят к утечке данных и переобучению, и данную проблему можно частично решить с применением mean target encoding на отложенной выборке, что работает заметно лучше, но такой способ не идеален так как остаётся меньше данных для обучения и вычисления статистик по категориальным признакам.

В CatBoost применяется модифицированный подход к mean target encoding, называемый Ordered target (CatBoost) encoding, который состоит из 3-х нижеперечисленных этапов.

Случайная перестановка

Образцы в наборе данных перемешиваются случайным образом, что позволяет создать более сбалансированный набор данных и снизить переобучение.

Статистики «по прошлому»

После случайной перестановки рассчитывается новое значение для текущего категориального признака следующим образом:

$CTR = \frac{TargetSum + Prior}{TotalCount + 1}$

Prior — константа, определяемая как среднее арифметическое целевых значений.

TargetSum — сумма целевых значений для конкретного категориального признака до текущего момента.

TotalCount — количество всех наблюдений для конкретного категориального признака до текущего момента.

Данная формула может быть применима как для регрессии, так и для мультиклассификации.

Пошаговый пример с нуля для мультиклассификации

!pip install category_encoders

from category_encoders.cat_boost import CatBoostEncoder

train = pd.DataFrame({"color": ["red", "blue", "black", "black", "blue", "blue", "red", "blue"]})
target = pd.Series([2, 1, 2, 1, 0, 3, 0, 1])

cbe_encoder = CatBoostEncoder()
train_cbe = cbe_encoder.fit_transform(train, target)
prior = target.mean()

print(f'prior: {prior}', '', sep='\n')
print(train_cbe)


prior: 1.25

    color
0  1.2500
1  1.2500
2  1.2500
3  1.6250
4  1.1250
5  0.7500
6  1.6250
7  1.3125

В случае c числовыми признаками создаётся набор меток с помощью квантования и производится их бинаризация на k классов: все значения до границы (метки) относятся к классу 0, а после к классу 1. Далее процедура схожа со случаем выше.

В CatBoost поддерживаются следующие виды квантования:

мода;
медианная сетка;
равномерная сетка + квантили (размер квантования уменьшается вдвое и разбиения (пороговые значения) объединяются по одному из двух способов: медиана или равномерная сетка);
MaxLogSum — максимизируется значение следующего выражения внутри каждого сегмента:

$\sum\limits_{i=1}^{n} log \ w_i$

MinEntropy — минимизируется значение следующего выражения внутри каждого сегмента:

$\sum\limits_{i=1}^{n} w_i \ log \ w_i$

GreedyLogSum — максимизируется аппроксимация жадным способом следующего выражения внутри каждого сегмента:

$\sum\limits_{i=1}^{n} log \ w_i$

где n — число различных объектов в сегменте, w (weight) — число повторений объекта в сегменте.

Комбинация категориальных признаков

В CatBoost используется комбинация категориальных признаков (например, красный диван, синий диван, синий стул и так далее), что позволяет модели учитывать неочевидные взаимосвязи между ними и улучшить качество итоговых прогнозов в целом. Для избегания экспоненциального роста комбинаций категориальных признаков, их создание происходит не перед началом обучения, а в момент построения каждого дерева жадным алгоритмом отдельно.

Структура решающих деревьев и упорядоченный бустинг

В качестве базовых моделей в CatBoost используются симметричные односторонние деревья, имеющие одинаковую глубину и ширину. Создатели алгоритма считают, что такой подход позволяет ускорить обучение и создавать деревья, которые менее подвержены переобучению.

Для нахождения наилучшего критерия разбиения в узле используется косинусное сходство — его максимальное значение будет соответствовать лучшему критерию:

$Cosine \ Similarity = \frac{ \sum\limits_{i=1}^{n} A_i B_i}{\sqrt{\sum\limits_{i=1}^{n} A_i^2} \sqrt{\sum\limits_{i=1}^{n} B_i^2}}$

где A — вектор с остаточными ошибками, B — вектор с прогнозами в листовых узлах.

Проблема классического бустинга заключается в том, что прогнозы базовых моделей могут иметь высокое смещение, что в конечном счёте приводит к переобучению модели в целом — это обусловлено тем, что обучение и оценка деревьев производится на одинаковом наборе данных (X_train). Данная проблема решается с помощью метода линейной аппроксимации упорядоченного бустинга (Ordered Boosting), суть которого заключается в следующем: данные перемешиваются случайным образом и после чего текущее дерево обучается на m-1 образцах с учётом остатков предыдущего, что очень похоже на принцип в ordered target encoding, то есть изначально берётся один образец из набора данных и модель учится на всех образцах до него, то есть на нулях, далее берётся два образца и модель учится на одном, после чего рассчитываются остатки, которые будут использованы для следующей модели и так далее. Такой подход позволяет произвести обучение дерева со сложностью O(log(m)) вместо O(m^2) .

Структура симметричного одностороннего дерева

Дополнительные возможности CatBoost

Работа с пропусками

Поддерживаются следующие режимы обработки пропусков:

"Forbidden" — пропуски интерпретируются как ошибки.
"Min" — пропуски обрабатываются как минимальное значение (меньшее, чем все остальные значения) для объекта.
"Max" — пропуски обрабатываются как максимальное значение (превышающее все остальные значения) для объекта.

Режим обработки по умолчанию — Min. Также при выборе деревьев учитывается разбиение, отделяющее пропущенные значения от всех остальных.

Snapshots

Данный параметр (save_snapshot=True) позволяет создавать резервные копии промежуточных результатов. Если происходит неожиданное прерывание обучения (например, случайное выключение компьютера), то оно может быть продолжено из сохранённого состояния и в таком случае не нужно повторять завершённые итерации построения деревьев.

Обучение на GPU

В CatBoost возможно обучение не только на нескольких GPU, но и сразу на нескольких серверах с GPU, что фактически позволяет запустить распределённое обучение и использовать датасеты любого размера. Данная процедура позволяет значительно ускорить работу алгоритма: чем больше датасет, тем больше будет прирост в производительности в сравнении с обучением на CPU. Для того чтобы этим воспользоваться достаточно указать task_type="GPU".

Пакетная реализация CatBoost и оценка полученных результатов

Как можно заметить из примера классификации, не смотря на большое количество улучшений, CatBoost далеко не всегда показывает высокие результаты в сравнении с другими реализациями градиентного бустинга, как это заявляют его разработчики, особенно заметна разница в случае с несбалансированными наборами данных. Однако, стоит отметить, что в случае хорошей подготовки данных, при прочих равных, в ряде случаев CatBoost будет отличаться от других модификаций немного более высокой точностью.

В случае регрессии, когда нет явно выраженной проблемы как в случае выше, CatBoost справляется довольно хорошо.

Полученные результаты приведены ниже.

!pip install catboost

CatBoostClassifier

from catboost import CatBoostClassifier, CatBoostRegressor

cb_clf = CatBoostClassifier(random_seed=0, logging_level='Silent')
cb_clf.fit(X1_train, y1_train)
cb_clf_pred_res = cb_clf.predict(X1_test).squeeze()   # remove all single-dimensional entries
cb_clf_accuracy = accuracy_score(cb_clf_pred_res, y1_test)
print(f'cb_clf accuracy: {cb_clf_accuracy}')
print(cb_clf_pred_res)

feature_indexes = [1, 3]
title3 = 'CatBoostClassifier surface'
decision_boundary_plot(X1, y1, X1_train, y1_train, cb_clf, feature_indexes, title3)


cb_clf accuracy: 0.6851851851851852
[5 0 1 4 1 1 0 1 1 1 0 0 1 1 1 5 2 1 2 2 1 0 5 1 0 0 5 0 1 1 0 0 1 0 0 0 0
 0 0 5 1 5 1 0 1 1 0 1 0 1 0 1 5 0]

CatBoostRegressor

cb_reg = CatBoostRegressor(random_seed=0, logging_level='Silent')
cb_reg.fit(X2_train, y2_train)
cb_reg_pred_res = cb_reg.predict(X2_test)
cb_reg_mape = mean_absolute_percentage_error(cb_reg_pred_res, y2_test)
print(f'cb_reg mape: {cb_reg_mape}')
print(cb_reg_pred_res)


cb_reg mape: 0.3260573594870776
[268.21698587 226.61112865 152.56164473 105.39534082 185.56084388
 231.13386917 112.94031582 246.11217136 121.83631454 238.66069067
 193.92810529 171.53207065 120.95427566  98.87431921 301.0159263
  97.989809   170.68360243  70.03044356 104.9534818  235.1624283
 174.7895105  131.39173163 178.28683674 128.13631321 230.60611298
 186.37090017 131.52230822  61.453026   231.95028492 148.67901568
 186.2317954   80.45908918 137.68617262 171.24445717 128.77648933
 178.31364898 179.17236177 149.82094832 102.25676058 209.48231728
 116.01443961 169.06621701 110.9047463  210.24240477 158.28102362
  72.4532758  135.86931928 121.14865532  76.8103627  278.66222581
 127.85077736  55.25704176 144.51194393 170.06468442 271.51905928
 187.61879537 159.56680649 118.65520621 121.92004997 183.56316238
 248.87847858 137.77180529 117.98743897  82.53109543 255.59388802
 151.23062005  94.00606836 257.15481144 227.21705762  96.77322241
  78.34198817 118.09701594  91.2069645   91.70192792 125.65538331
 155.68222799 129.63411563 230.46496307 292.38275237 213.68057983
 131.76653094 217.93105223  50.37188156 256.16334997  94.4107044
  90.95547399 139.58356594 187.26517    127.97009716 171.85831366
  89.65051176  95.93129442  60.14437246 178.27466698 101.36657382
 106.15753799 206.11327567 217.74079838 130.83283251 138.63144499
 154.18156801 123.38315127 197.65233246 125.95249034 239.25024631
 139.69849648 245.02582928 288.17892899 122.76437963  70.023686
 233.36946028]

Преимущества и недостатки CatBoost

Преимущества:

высокая точность и скорость обучения в большинстве случаев;
возможность работы с категориальными признаками;
большое количество гиперпараметров для настройки алгоритма;
применяется для многих задач.

Недостатки:

нестабильная работа с несбалансированными данными;
сложность в конфигурации из-за большого количества гиперпараметров;
плохая документация (личное мнение).

Дополнительные источники

Статья «CatBoost: unbiased boosting with categorical features», Liudmila Prokhorenkova, Gleb Gusev, Aleksandr Vorobev, Anna Veronika Dorogush, Andrey Gulin.

Документация.

Лекция.

Пошаговое построение CatBoost (в ordered target encoding есть ошибки):

один;
два.

Особенности LightGBM

LightGBM (Light Gradient Boosting Machine) — реализация градиентного бустинга от компании Microsoft, способная обрабатывать большие наборы данных и при этом обеспечивать высокую точность. Рассмотрим более подробно ключевые особенности, которые на это повлияли.

Gradient-based One-Side Sampling (GOSS)

В отличие от классического градиентного бустинга, где регрессионные деревья обучаются на всём множестве X_train, в LightGBM деревья обучаются на подмножестве данных, полученном с помощью одностороннего отбора на основе градиента (GOSS), суть которого заключается в сортировке и выборе образцов top N с наибольшим градиентом — это означает, что именно этим данным следует уделить особое внимание, поскольку на них дерево будет ошибаться больше всего. Помимо этого, случайным образом производится отбор части образцов с низким градиентом с последующим присвоением им весов $\frac{1-a}{b}$ , чтобы избежать изменений в распределении данных. В данном случае — доля образцов top N в процентах с наибольшим градиентом, — доля образцов в процентах с низким градиентом.

Проще говоря, использование GOSS позволяет значительно ускорить обучение каждого последующего дерева в ансамбле путём уменьшения обучающего набора данных без особой потери в качестве прогнозов в большинстве случаев.

Histogram-based tree node splitting

При поиске наилучшего критерия разбиения узла в LightGBM также можно использовать гистограмму значений признаков: перебираются все пороговые значения гистограммы вместо всех значений признаков, что позволяет существенно ускорить поиск наилучшего критерия. Например, для признака "рост" можно использовать вопросы "рост" ⩽ 160, "рост" ⩽ 170 вместо "рост" ⩽ 163, "рост" ⩽ 167 и так далее. Как можно было заметить, гистограммы применяются в том или ином виде во всех трёх модификациях градиентного бустинга.

Exclusive Feature Bundling (EFB)

Предположим, что тем или иным образом в датасете имеются закодированные категориальные признаки, большинство из которых являются взаимоисключающими (те, которые одновременно не принимают ненулевые значения). При большом числе таких признаков обучение модели может занять слишком много времени и для решения данной проблемы в LightGBM применяется понижение размерности пространства признаков путём объединения взаимоисключающих признаков (EFB).

Стоит отметить, что в действительности строго взаимоисключающие признаки в данных встречаются крайне редко. Поэтому вводится такое понятие как доля конфликтов между признаками — отношение наблюдений от общего числа, где два признака принимают одновременно ненулевые значения.

Допустим, имеется датасет с признаками F0−F4 . В первую очередь стоит установить допустимую долю конфликтов, например, 25%. Тогда в случае из 8 образцов допустимое число конфликтов будет составлять 2. Далее строится матрица конфликтов между всеми признаками и производится подсчёт общей суммы конфликтов для каждого признака, после чего происходит их сортировка в порядке убывания.

На основе матрицы конфликтов строится взвешенный граф, у которого в полученном порядке отсекаются все рёбра с числом конфликтов, превышающим допустимое значение.

Работа Exclusive Feature Bundling с нуля на простом примере

Первой связкой будет F_0−F_1 . Не смотря на то что F_1−F_2 и F_1−F_3 не конфликтуют, их объединение невозможно из-за конфликтов F_0−F_2 и F_0−F_3 . Следующей связкой будет F_2−F_4 . Не смотря на то, что F_4 и F_3 не конфликтуют, их объединение невозможно из-за конфликта F_2−F_3 . Последним остаётся признак F_3 .

Новая связка будет состоять из ненулевых значений первого признака и ненулевых значений второго плюс смещение, равное максимальному значению первого признака: смещение каждого следующего признака будет равно сумме максимальных значений предыдущих. В случаях, когда наблюдается конфликт, берётся значение последнего признака со смещением.

Подробные расчёты приведены ниже.

Нетрудно заметить, что алгоритм EFB можно оптимизировать, игнорируя вычисления для всех нулевых значений в признаках. Также данную оптимизацию можно применить к histogram-based tree node splitting, что в целом снизит сложность построения гистограммы с O(data) до O(non-zero data).

Leaf-wise tree growth

Вместо построения деревьев слой за слоем, в LightGBM деревья строятся в глубину: рост продолжается в листовом узле с наибольшим информационным приростом, в результате чего дерево получается глубоким и несимметричным. Чаще всего такой подход превосходит симметричные деревья в плане точности.

Обучение на GPU

При работе с большими датасетами в LightGBM поддерживается обучение не только на GPU, но и распределённое обучение сразу на нескольких машинах.

На данный момент существуют следующие подходы:

Data parallel (образцы датасета разбиваются на локальные гистограммы, каждая из которых обрабатывается отдельно);
Feature parallel (признаки датасета разбиваются на локальные гистограммы, каждая из которых обрабатывается отдельно);
Voting parallel (на каждом процессе локально выбирается K лучших признаков для разбиения, после чего выбирается наилучший из них глобально).

Пакетная реализация LightGBM и оценка полученных результатов

Как можно заметить, в случае классификации LightGBM показал результат из коробки значительно лучше, чем предыдущие модификации бустинга. Также стоит отметить, что из всех трёх рассмотренных модификацией градиентного бустинга LightGBM является самой быстрой: чем больше датасет, тем сильнее будет заметна разница в плане скорости.

Полученные результаты приведены ниже.

LGBMClassifier

from lightgbm import LGBMClassifier, LGBMRegressor

lgb_clf = LGBMClassifier(random_state=0, verbose=-1)
lgb_clf.fit(X1_train, y1_train)
lgb_clf_pred_res = lgb_clf.predict(X1_test)
lgb_clf_accuracy = accuracy_score(lgb_clf_pred_res, y1_test)
print(f'lgb_clf accuracy: {lgb_clf_accuracy}')
print(lgb_clf_pred_res)

feature_indexes = [1, 3]
title4 = 'LGBMClassifier surface'
decision_boundary_plot(X1, y1, X1_train, y1_train, lgb_clf, feature_indexes, title4)


lgb_clf accuracy: 0.7407407407407407
[5 0 1 4 1 1 0 1 1 1 0 0 0 1 1 5 2 1 2 2 3 1 5 5 0 0 5 0 1 1 0 0 1 0 0 0 0
 0 0 5 1 4 1 0 0 1 0 1 0 1 0 1 5 0]

LGBMRegressor

lgb_reg = LGBMRegressor(random_state=0, verbose=-1)
lgb_reg.fit(X2_train, y2_train)
lgb_reg_pred_res = lgb_reg.predict(X2_test)
lgb_reg_mape = mean_absolute_percentage_error(lgb_reg_pred_res, y2_test)
print(f'lgb_reg_mape: {lgb_reg_mape}')
print(lgb_reg_pred_res)


lgb_reg_mape: 0.349719771087187
[261.46864967 223.24674049 179.60231224 104.76984808 197.67529058
 264.26116883 111.11355079 243.72059654 144.18045853 239.08131542
 199.90841705 170.37417433 151.67933516  89.30779664 297.93561211
  96.58862229 162.53285879  70.06113305 108.36436227 234.16276627
 207.7552967  136.83913826 172.38724529 110.26981866 228.1574526
 163.56817258 108.94516511  73.93303203 233.31869769 116.00042252
 195.39846082  72.54403358 164.38943926 194.42418913 130.42154381
 183.04435501 173.56352142 147.00774306  89.07365731 175.25759644
 108.72749406 144.48000593  99.60713604 196.86889512 160.47835096
  60.17798032 133.4909559  115.46543242  64.48286456 251.20520649
 135.52608509  45.16396653 161.70420785 187.28406745 274.21158331
 179.30355071 150.1563969  111.1275583  100.5067047  133.7868607
 269.98464504 159.03980859 118.01578638  75.41625416 249.0569831
 133.99026562 111.43954756 221.81954307 206.02676942  81.56098377
  72.5920367  101.31664062  93.31261471 111.24487763 149.40048547
 177.37205369 130.08118649 212.0573     294.138509   232.24330644
 109.5457663  181.3487432   63.20386576 261.47508264  88.43007908
  99.59568927 127.97978786 211.77672798 121.44351586 138.27282283
 121.01455597  91.26965383  68.92870597 156.03234199 101.80734511
 108.60244399 214.13968395 218.58499151 131.09438025 114.99269899
 187.63688894 117.5399017  165.85959666 110.74981366 238.97897095
 114.01806217 240.07138204 283.0195311  108.08395213  67.44250127
 238.97271264]

Преимущества и недостатки LightGBM

Преимущества:

высокая точность прогнозов и особенно скорость обучения;
возможность работы с категориальными признаками;
большое количество гиперпараметров для настройки алгоритма;
применяется для многих задач.

Недостатки:

возможна нестабильная работа на датасетах небольшого размера;
сложность в конфигурации из-за большого количества гиперпараметров.

Дополнительные источники

Статья «LightGBM: A Highly Efficient Gradient Boosting Decision Tree», Guolin Ke, Qi Meng, Thomas Finley, Taifeng Wang, Wei Chen, Weidong Ma, Qiwei Ye, Tie-Yan Liu.

Документация.

Лекции:

Какой бустинг лучше

Однозначного ответа на вопрос какой бустинг лучше нет. Если раньше можно было сказать, что при работе с категориальными признаками лучше использовать CatBoost, когда важна скорость — LightGBM, ну или XGBoost, если хочется стабильное решение и нет времени на эксперименты, то сейчас такое утверждение будет неверным: как можно было заметить, все три модификации успели скопировать друг у друга много хороших идей, но исторически они отличались довольно сильно.

Более правильным будет утверждение, что выбор наилучшего бустинга будет зависеть от конкретной задачи и данных, которые применяются для обучения модели: в разных сценариях одни и те же алгоритмы могут показывать как лучшие результаты в плане скорости и точности, так и наоборот — худшие. Таким образом, выбор наиболее оптимального бустинга будет заключаться в проведении экспериментов с разными модификациями и сравнении полученных результатов.

Несколько полезных источников со сравнением бустингов: один, два, три.

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13 мар. 2024 г.

07.09.23 16:24 CherryTeam

Cherry Team atlyginimų skaičiavimo programa yra labai naudingas įrankis įmonėms, kai reikia efektyviai valdyti ir skaičiuoti darbuotojų atlyginimus. Ši programinė įranga, turinti išsamias funkcijas ir patogią naudotojo sąsają, suteikia daug privalumų, kurie padeda supaprastinti darbo užmokesčio skaičiavimo procesus ir pagerinti finansų valdymą. Štai keletas pagrindinių priežasčių, kodėl Cherry Team atlyginimų skaičiavimo programa yra naudinga įmonėms: Automatizuoti ir tikslūs skaičiavimai: Atlyginimų skaičiavimai rankiniu būdu gali būti klaidingi ir reikalauti daug laiko. Programinė įranga Cherry Team automatizuoja visą atlyginimų skaičiavimo procesą, todėl nebereikia atlikti skaičiavimų rankiniu būdu ir sumažėja klaidų rizika. Tiksliai apskaičiuodama atlyginimus, įskaitant tokius veiksnius, kaip pagrindinis atlyginimas, viršvalandžiai, premijos, išskaitos ir mokesčiai, programa užtikrina tikslius ir be klaidų darbo užmokesčio skaičiavimo rezultatus. Sutaupoma laiko ir išlaidų: Darbo užmokesčio valdymas gali būti daug darbo jėgos reikalaujanti užduotis, reikalaujanti daug laiko ir išteklių. Programa Cherry Team supaprastina ir pagreitina darbo užmokesčio skaičiavimo procesą, nes automatizuoja skaičiavimus, generuoja darbo užmokesčio žiniaraščius ir tvarko išskaičiuojamus mokesčius. Šis automatizavimas padeda įmonėms sutaupyti daug laiko ir pastangų, todėl žmogiškųjų išteklių ir finansų komandos gali sutelkti dėmesį į strategiškai svarbesnę veiklą. Be to, racionalizuodamos darbo užmokesčio operacijas, įmonės gali sumažinti administracines išlaidas, susijusias su rankiniu darbo užmokesčio tvarkymu. Mokesčių ir darbo teisės aktų laikymasis: Įmonėms labai svarbu laikytis mokesčių ir darbo teisės aktų, kad išvengtų baudų ir teisinių problemų. Programinė įranga Cherry Team seka besikeičiančius mokesčių įstatymus ir darbo reglamentus, užtikrindama tikslius skaičiavimus ir teisinių reikalavimų laikymąsi. Programa gali dirbti su sudėtingais mokesčių scenarijais, pavyzdžiui, keliomis mokesčių grupėmis ir įvairių rūšių atskaitymais, todėl užtikrina atitiktį reikalavimams ir kartu sumažina klaidų riziką. Ataskaitų rengimas ir analizė: Programa Cherry Team siūlo patikimas ataskaitų teikimo ir analizės galimybes, suteikiančias įmonėms vertingų įžvalgų apie darbo užmokesčio duomenis. Ji gali generuoti ataskaitas apie įvairius aspektus, pavyzdžiui, darbo užmokesčio paskirstymą, išskaičiuojamus mokesčius ir darbo sąnaudas. Šios ataskaitos leidžia įmonėms analizuoti darbo užmokesčio tendencijas, nustatyti tobulintinas sritis ir priimti pagrįstus finansinius sprendimus. Pasinaudodamos duomenimis pagrįstomis įžvalgomis, įmonės gali optimizuoti savo darbo užmokesčio strategijas ir veiksmingai kontroliuoti išlaidas. Integracija su kitomis sistemomis: Cherry Team programinė įranga dažnai sklandžiai integruojama su kitomis personalo ir apskaitos sistemomis. Tokia integracija leidžia automatiškai perkelti atitinkamus duomenis, pavyzdžiui, informaciją apie darbuotojus ir finansinius įrašus, todėl nebereikia dubliuoti duomenų. Supaprastintas duomenų srautas tarp sistemų padidina bendrą efektyvumą ir sumažina duomenų klaidų ar neatitikimų riziką. Cherry Team atlyginimų apskaičiavimo programa įmonėms teikia didelę naudą - automatiniai ir tikslūs skaičiavimai, laiko ir sąnaudų taupymas, atitiktis mokesčių ir darbo teisės aktų reikalavimams, ataskaitų teikimo ir analizės galimybės bei integracija su kitomis sistemomis. Naudodamos šią programinę įrangą įmonės gali supaprastinti darbo užmokesčio skaičiavimo procesus, užtikrinti tikslumą ir atitiktį reikalavimams, padidinti darbuotojų pasitenkinimą ir gauti vertingų įžvalgų apie savo finansinius duomenis. Programa Cherry Team pasirodo esanti nepakeičiamas įrankis įmonėms, siekiančioms efektyviai ir veiksmingai valdyti darbo užmokestį. https://cherryteam.lt/lt/
08.10.23 01:30 davec8080

The "Shibarium for this confirmed rug pull is a BEP-20 project not related at all to Shibarium, SHIB, BONE or LEASH. The Plot Thickens. Someone posted the actual transactions!!!! https://bscscan.com/tx/0xa846ea0367c89c3f0bbfcc221cceea4c90d8f56ead2eb479d4cee41c75e02c97 It seems the article is true!!!! And it's also FUD. Let me explain. Check this link: https://bscscan.com/token/0x5a752c9fe3520522ea88f37a41c3ddd97c022c2f So there really is a "Shibarium" token. And somebody did a rug pull with it. CONFIRMED. But the "Shibarium" token for this confirmed rug pull is a BEP-20 project not related at all to Shibarium, SHIB, BONE or LEASH.
24.06.24 04:31 tashandiarisha

Web-site. https://trustgeekshackexpert.com/ Tele-Gram, trustgeekshackexpert During the pandemic, I ventured into the world of cryptocurrency trading. My father loaned me $10,000, which I used to purchase my first bitcoins. With diligent research and some luck, I managed to grow my investment to over $350,000 in just a couple of years. I was thrilled with my success, but my excitement was short-lived when I decided to switch brokers and inadvertently fell victim to a phishing attack. While creating a new account, I received what seemed like a legitimate email requesting verification. Without second-guessing, I provided my information, only to realize later that I had lost access to my email and cryptocurrency wallets. Panic set in as I watched my hard-earned assets disappear before my eyes. Desperate to recover my funds, I scoured the internet for solutions. That's when I stumbled upon the Trust Geeks Hack Expert on the Internet. The service claimed to specialize in recovering lost crypto assets, and I decided to take a chance. Upon contacting them, the team swung into action immediately. They guided me through the entire recovery process with professionalism and efficiency. The advantages of using the Trust Geeks Hack Expert Tool became apparent from the start. Their team was knowledgeable and empathetic, understanding the urgency and stress of my situation. They employed advanced security measures to ensure my information was handled safely and securely. One of the key benefits of the Trust Geeks Hack Expert Tool was its user-friendly interface, which made a complex process much more manageable for someone like me, who isn't particularly tech-savvy. They also offered 24/7 support, so I never felt alone during recovery. Their transparent communication and regular updates kept me informed and reassured throughout. The Trust Geeks Hack Expert Tool is the best solution for anyone facing similar issues. Their swift response, expertise, and customer-centric approach set them apart from other recovery services. Thanks to their efforts, I regained access to my accounts and my substantial crypto assets. The experience taught me a valuable lesson about online security and showed me the incredible potential of the Trust Geeks Hack Expert Tool. Email:: trustgeekshackexpert{@}fastservice{.}com WhatsApp + 1.7.1.9.4.9.2.2.6.9.3
26.06.24 18:46 Jacobethannn098

LEGAL RECOUP FOR CRYPTO THEFT BY ADRIAN LAMO HACKER
26.06.24 18:46 Jacobethannn098

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04.07.24 04:49 ZionNaomi

For over twenty years, I've dedicated myself to the dynamic world of marketing, constantly seeking innovative strategies to elevate brand visibility in an ever-evolving landscape. So when the meteoric rise of Bitcoin captured my attention as a potential avenue for investment diversification, I seized the opportunity, allocating $20,000 to the digital currency. Witnessing my investment burgeon to an impressive $70,000 over time instilled in me a sense of financial promise and stability.However, amidst the euphoria of financial growth, a sudden and unforeseen oversight brought me crashing back to reality during a critical business trip—I had misplaced my hardware wallet. The realization that I had lost access to the cornerstone of my financial security struck me with profound dismay. Desperate for a solution, I turned to the expertise of Daniel Meuli Web Recovery.Their response was swift . With meticulous precision, they embarked on the intricate process of retracing the elusive path of my lost funds. Through their unwavering dedication, they managed to recover a substantial portion of my investment, offering a glimmer of hope amidst the shadows of uncertainty. The support provided by Daniel Meuli Web Recovery extended beyond mere financial restitution. Recognizing the imperative of fortifying against future vulnerabilities, they generously shared invaluable insights on securing digital assets. Their guidance encompassed crucial aspects such as implementing hardware wallet backups and fortifying security protocols, equipping me with recovered funds and newfound knowledge to navigate the digital landscape securely.In retrospect, this experience served as a poignant reminder of the critical importance of diligence and preparedness in safeguarding one's assets. Thanks to the expertise and unwavering support extended by Daniel Meuli Web Recovery, I emerged from the ordeal with renewed resilience and vigilance. Empowered by their guidance and fortified by enhanced security measures, I now approach the future with unwavering confidence.The heights of financial promise to the depths of loss and back again has been a humbling one, underscoring the volatility and unpredictability inherent in the digital realm. Yet, through adversity, I have emerged stronger, armed with a newfound appreciation for the importance of diligence, preparedness, and the invaluable support of experts like Daniel Meuli Web Recovery.As I persist in traversing the digital landscape, I do so with a judicious blend of vigilance and fortitude, cognizant that with adequate safeguards and the backing of reliable confidants, I possess the fortitude to withstand any adversity that may arise. For this, I remain eternally appreciative. Email Danielmeuliweberecovery @ email . c om WhatsApp + 393 512 013 528
13.07.24 21:13 michaelharrell825

In 2020, amidst the economic fallout of the pandemic, I found myself unexpectedly unemployed and turned to Forex trading in hopes of stabilizing my finances. Like many, I was drawn in by the promise of quick returns offered by various Forex robots, signals, and trading advisers. However, most of these products turned out to be disappointing, with claims that were far from reality. Looking back, I realize I should have been more cautious, but the allure of financial security clouded my judgment during those uncertain times. Amidst these disappointments, Profit Forex emerged as a standout. Not only did they provide reliable service, but they also delivered tangible results—a rarity in an industry often plagued by exaggerated claims. The positive reviews from other users validated my own experience, highlighting their commitment to delivering genuine outcomes and emphasizing sound financial practices. My journey with Profit Forex led to a net profit of $11,500, a significant achievement given the challenges I faced. However, my optimism was short-lived when I encountered obstacles trying to withdraw funds from my trading account. Despite repeated attempts, I found myself unable to access my money, leaving me frustrated and uncertain about my financial future. Fortunately, my fortunes changed when I discovered PRO WIZARD GIlBERT RECOVERY. Their reputation for recovering funds from fraudulent schemes gave me hope in reclaiming what was rightfully mine. With a mixture of desperation and cautious optimism, I reached out to them for assistance. PRO WIZARD GIlBERT RECOVERY impressed me from the start with their professionalism and deep understanding of financial disputes. They took a methodical approach, using advanced techniques to track down the scammers responsible for withholding my funds. Throughout the process, their communication was clear and reassuring, providing much-needed support during a stressful period. Thanks to PRO WIZARD GIlBERT RECOVERY's expertise and unwavering dedication, I finally achieved a resolution to my ordeal. They successfully traced and retrieved my funds, restoring a sense of justice and relief. Their intervention not only recovered my money but also renewed my faith in ethical financial services. Reflecting on my experience, I've learned invaluable lessons about the importance of due diligence and discernment in navigating the Forex market. While setbacks are inevitable, partnering with reputable recovery specialists like PRO WIZARD GIlBERT RECOVERY can make a profound difference. Their integrity and effectiveness have left an indelible mark on me, guiding my future decisions and reinforcing the value of trustworthy partnerships in achieving financial goals. I wholeheartedly recommend PRO WIZARD GIlBERT RECOVERY to anyone grappling with financial fraud or disputes. Their expertise and commitment to client satisfaction are unparalleled, offering a beacon of hope in challenging times. Thank you, PRO WIZARD GIlBERT RECOVERY, for your invaluable assistance in reclaiming what was rightfully mine. Your service not only recovered my funds but also restored my confidence in navigating the complexities of financial markets with greater caution and awareness. Email: prowizardgilbertrecovery(@)engineer.com Homepage: https://prowizardgilbertrecovery.xyz WhatsApp: +1 (516) 347‑9592
17.07.24 02:26 thompsonrickey

In the vast and often treacherous realm of online investments, I was entangled in a web of deceit that cost me nearly $45,000. It all started innocuously enough with an enticing Instagram profile promising lucrative returns through cryptocurrency investment. Initially, everything seemed promising—communications were smooth, and assurances were plentiful. However, as time passed, my optimism turned to suspicion. Withdrawal requests were met with delays and excuses. The once-responsive "investor" vanished into thin air, leaving me stranded with dwindling hopes and a sinking feeling in my gut. It became painfully clear that I had been duped by a sophisticated scheme designed to exploit trust and naivety. Desperate to recover my funds, I turned to online forums where I discovered numerous testimonials advocating for Muyern Trust Hacker. With nothing to lose, I contacted them, recounting my ordeal with a mixture of skepticism and hope. Their swift response and professional demeanor immediately reassured me that I had found a lifeline amidst the chaos. Muyern Trust Hacker wasted no time in taking action. They meticulously gathered evidence, navigated legal complexities, and deployed their expertise to expedite recovery. In what felt like a whirlwind of activity, although the passage of time was a blur amidst my anxiety, they achieved the seemingly impossible—my stolen funds were returned. The relief I felt was overwhelming. Muyern Trust Hacker not only restored my financial losses but also restored my faith in justice. Their commitment to integrity and their relentless pursuit of resolution were nothing short of remarkable. They proved themselves as recovery specialists and guardians against digital fraud, offering hope to victims like me who had been ensnared by deception. My gratitude knows no bounds for Muyern Trust Hacker. Reach them at muyerntrusted @ m a i l - m e . c o m AND Tele gram @ muyerntrusthackertech
18.07.24 20:13 austinagastya

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27.08.24 12:50 James889900

All you need is to hire an expert to help you accomplish that. If there’s any need to spy on your partner’s phone. From my experience I lacked evidence to confront my husband on my suspicion on his infidelity, until I came across ETHICALAHCKERS which many commend him of assisting them in their spying mission. So I contacted him and he provided me with access into his phone to view all text messages, call logs, WhatsApp messages and even her location. This evidence helped me move him off my life . I recommend you consult ETHICALHACKERS009 @ gmail.com OR CALL/TEXT ‪+1(716) 318-5536 or whatsapp +14106350697 if you need access to your partner’s phone
27.08.24 13:06 James889900

All you need is to hire an expert to help you accomplish that. If there’s any need to spy on your partner’s phone. From my experience I lacked evidence to confront my husband on my suspicion on his infidelity, until I came across ETHICALAHCKERS which many commend him of assisting them in their spying mission. So I contacted him and he provided me with access into his phone to view all text messages, call logs, WhatsApp messages and even her location. This evidence helped me move him off my life . I recommend you consult ETHICALHACKERS009 @ gmail.com OR CALL/TEXT ‪+1(716) 318-5536 or whatsapp +14106350697 if you need access to your partner’s phone
02.09.24 20:24 [email protected]

If You Need Hacker To Recover Your Bitcoin Contact Paradox Recovery Wizard Paradox Recovery Wizard successfully recovered $123,000 worth of Bitcoin for my husband, which he had lost due to a security breach. The process was efficient and secure, with their expert team guiding us through each step. They were able to trace and retrieve the lost cryptocurrency, restoring our peace of mind and financial stability. Their professionalism and expertise were instrumental in recovering our assets, and we are incredibly grateful for their service. Email: support@ paradoxrecoverywizard.com Email: paradox_recovery @cyberservices.com Wep: https://paradoxrecoverywizard.com/ WhatsApp: +39 351 222 3051.
06.09.24 01:35 Celinagarcia

HOW TO RECOVER MONEY LOST IN BITCOIN/USDT TRADING OR TO CRYPTO INVESTMENT !! Hi all, friends and families. I am writing From Alberton Canada. Last year I tried to invest in cryptocurrency trading in 2023, but lost a significant amount of money to scammers. I was cheated of my money, but thank God, I was referred to Hack Recovery Wizard they are among the best bitcoin recovery specialists on the planet. they helped me get every penny I lost to the scammers back to me with their forensic techniques. and I would like to take this opportunity to advise everyone to avoid making cryptocurrency investments online. If you have already lost money on forex, cryptocurrency or Ponzi schemes, please contact [email protected] or WhatsApp: +1 (757) 237–1724 at once they can help you get back the crypto you lost to scammers. BEST WISHES. Celina Garcia.
06.09.24 01:44 Celinagarcia

HOW TO RECOVER MONEY LOST IN BITCOIN/USDT TRADING OR TO CRYPTO INVESTMENT !! Hi all, friends and families. I am writing From Alberton Canada. Last year I tried to invest in cryptocurrency trading in 2023, but lost a significant amount of money to scammers. I was cheated of my money, but thank God, I was referred to Hack Recovery Wizard they are among the best bitcoin recovery specialists on the planet. they helped me get every penny I lost to the scammers back to me with their forensic techniques. and I would like to take this opportunity to advise everyone to avoid making cryptocurrency investments online. If you have already lost money on forex, cryptocurrency or Ponzi schemes, please contact [email protected] or WhatsApp: +1 (757) 237–1724 at once they can help you get back the crypto you lost to scammers. BEST WISHES. Celina Garcia.
16.09.24 00:10 marcusaustin

Bitcoin Recovery Services: Restoring Lost Cryptocurrency If you've lost access to your cryptocurrency and unable to make a withdrawal, I highly recommend iBolt Cyber Hacker Bitcoin Recovery Services. Their team is skilled, professional, and efficient in recovering lost Bitcoin. They provide clear communication, maintain high security standards, and work quickly to resolve issues. Facing the stress of lost cryptocurrency, iBolt Cyber Hacker is a trusted service that will help you regain access to your funds securely and reliably. Highly recommended! Email: S u p p o r t @ ibolt cyber hack . com Cont/Whtp + 3. .9 .3. .5..0. .9. 2. 9. .0 .3. 1 .8. Website: h t t p s : / / ibolt cyber hack . com /
16.09.24 00:11 marcusaustin

Bitcoin Recovery Services: Restoring Lost Cryptocurrency If you've lost access to your cryptocurrency and unable to make a withdrawal, I highly recommend iBolt Cyber Hacker Bitcoin Recovery Services. Their team is skilled, professional, and efficient in recovering lost Bitcoin. They provide clear communication, maintain high security standards, and work quickly to resolve issues. Facing the stress of lost cryptocurrency, iBolt Cyber Hacker is a trusted service that will help you regain access to your funds securely and reliably. Highly recommended! Email: S u p p o r t @ ibolt cyber hack . com Cont/Whtp + 3. .9 .3. .5..0. .9. 2. 9. .0 .3. 1 .8. Website: h t t p s : / / ibolt cyber hack . com /
23.09.24 18:56 matthewshimself

At first, I was admittedly skeptical about Worldcoin (ref: https://worldcoin.org/blog/worldcoin/this-is-worldcoin-video-explainer-series), particularly around the use of biometric data and the WLD token as a reward mechanism for it. However, after following the project closer, I’ve come to appreciate the broader vision and see the value in the underlying tech behind it. The concept of Proof of Personhood (ref: https://worldcoin.org/blog/worldcoin/proof-of-personhood-what-it-is-why-its-needed) has definitely caught my attention, and does seem like a crucial step towards tackling growing issues like bots, deepfakes, and identity fraud. Sam Altman’s vision is nothing short of ambitious, but I do think he & Alex Blania have the chops to realize it as mainstay in the global economy.
01.10.24 14:54 Sinewclaudia

I lost about $876k few months ago trading on a fake binary option investment websites. I didn't knew they were fake until I tried to withdraw. Immediately, I realized these guys were fake. I contacted Sinew Claudia world recovery, my friend who has such experience before and was able to recover them, recommended me to contact them. I'm a living testimony of a successful recovery now. You can contact the legitimate recovery company below for help and assistance. [email protected] [email protected] WhatsApp: 6262645164
02.10.24 22:27 Emily Hunter

Can those who have fallen victim to fraud get their money back? Yes, you might be able to get back what was taken from you if you fell prey to a fraud from an unregulated investing platform or any other scam, but only if you report it to the relevant authorities. With the right plan and supporting documentation, you can get back what you've lost. Most likely, the individuals in control of these unregulated platforms would attempt to convince you that what happened to your money was a sad accident when, in fact, it was a highly skilled heist. You should be aware that there are resources out there to help you if you or someone you know has experienced one of these circumstances. Do a search using (deftrecoup (.) c o m). Do not let the perpetrators of this hoaxes get away with ruining you mentally and financially.
18.10.24 09:34 freidatollerud

The growth of WIN44 in Brazil is very interesting! If you're looking for more options for online betting and casino games, I recommend checking out Casinos in Brazil. It's a reliable platform that offers a wide variety of games and provides a safe and enjoyable experience for users. It's worth checking out! https://win44.vip
31.10.24 00:13 ytre89

Can those who have fallen victim to fraud get their money back? Yes, you might be able to get back what was taken from you if you fell prey to a fraud from an unregulated investing platform or any other scam, but only if you report it to the relevant authorities. With the right plan and supporting documentation, you can get back what you've lost. Most likely, the individuals in control of these unregulated platforms would attempt to convince you that what happened to your money was a sad accident when, in fact, it was a highly skilled heist. You should be aware that there are resources out there to help you if you or someone you know has experienced one of these circumstances. Do a search using (deftrecoup (.) c o m). Do not let the perpetrators of this hoaxes get away with ruining you mentally and financially.
02.11.24 14:44 diannamendoza732

In the world of Bitcoin recovery, Pro Wizard Gilbert truly represents the gold standard. My experience with Gilbert revealed just how exceptional his methods are and why he stands out as the premier authority in this critical field. When I first encountered the complexities of Bitcoin recovery, I was daunted by the technical challenges and potential risks. Gilbert’s approach immediately distinguished itself through its precision and effectiveness. His methods are meticulously designed, combining cutting-edge techniques with an in-depth understanding of the Bitcoin ecosystem. He tackled the recovery process with a level of expertise and thoroughness that was both impressive and reassuring. What sets Gilbert’s methods apart is not just their technical sophistication but also their strategic depth. He conducts a comprehensive analysis of each case, tailoring his approach to address the unique aspects of the situation. This personalized strategy ensures that every recovery effort is optimized for success. Gilbert’s transparent communication throughout the process was invaluable, providing clarity and confidence during each stage of the recovery. The results I achieved with Pro Wizard Gilbert’s methods were remarkable. His gold standard approach not only recovered my Bitcoin but did so with an efficiency and reliability that exceeded my expectations. His deep knowledge, innovative techniques, and unwavering commitment make him the definitive expert in Bitcoin recovery. For anyone seeking a benchmark in Bitcoin recovery solutions, Pro Wizard Gilbert’s methods are the epitome of excellence. His ability to blend technical prowess with strategic insight truly sets him apart in the industry. Call: for help. You may get in touch with them at ; Email: (prowizardgilbertrecovery(@)engineer.com) Telegram ; https://t.me/Pro_Wizard_Gilbert_Recovery Homepage ; https://prowizardgilbertrecovery.info
12.11.24 00:50 TERESA

Brigadia Tech Remikeable recovery has exhibited unparalleled strength in the realm of recovery. They stand out as the premier team to collaborate with if you encounter withdrawal difficulties from the platform where you’ve invested. Recently, I engaged with them to recover over a million dollars trapped in an investment platform I’d been involved with for months. I furnished their team with every detail of the investment, including accounts, names, and wallet addresses to which I sent the funds. This decision proved to be the best I’ve made, especially after realizing the company had scammed me. Brigadia Tech Remikeable recovery ensures exemplary service delivery and ensures the perpetrators face justice. They employ advanced techniques to ensure you regain access to your funds. Understandably, many individuals who have fallen victim to investment scams may still regret engaging in online services again due to the trauma of being scammed. However, I implore you to take action. Seek assistance from Brigadia Tech Remikeable Recovery Experts today and witness their remarkable capabilities. I am grateful that I resisted their enticements, and despite the time it took me to discover brigadia tech recovery, they ultimately fulfilled my primary objective. Without Brigadia Tech Recovery's intervention, I would have remained despondent and perplexed indefinitely. Also if you are looking for the best and safest investment company you can contact them, for wallet recovery, difficult withdrawal, etc. I am so happy to keep getting my daily BTC, all I do is keep 0.1 BTC in my mining wallet with the help of Brigadia Tech. They connected me to his mining stream and I earn 0.4 btc per day with this, my daily profit. I can get myself a new house and car. I can’t believe I have thousands of dollars in my bank account. Now you can get in. ([email protected]) Telegram +1 (323)-9 1 0 -1 6 0 5
17.11.24 09:31 Vivianlocke223

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19.11.24 03:06 [email protected]

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19.11.24 03:07 [email protected]

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21.11.24 08:02 Emily Hunter

If I hadn't found a review online and filed a complaint via email to support@deftrecoup. com , the people behind this unregulated scheme would have gotten away with leaving me in financial ruins. It was truly the most difficult period of my life.
22.11.24 04:41 [email protected]

I never could have imagined the nightmare of losing access to my digital wallet. All of my cryptocurrency holdings were abruptly imprisoned, inaccessible, and appeared to be lost forever following a catastrophic hardware breakdown. Years of meticulous investment and careful saving were reduced to nothing more than strings of code that I could no longer control, and I could feel the dread and sorrow that swept through me at that very instant. Thankfully, during my worst moment, I came into (TRUST GEEKS HACK EXPERT), a professional service devoted to recovering lost or inaccessible digital data. With optimism, I went out to their team of skilled technologists, laying bare the full nature of my issue. What followed was a laborious, multi-step process that required an almost surgical level of digital forensics and Bitcoin skill. In order to create a thorough profile of my wallet's contents and activities, the (TRUST GEEKS HACK EXPERT) team first thoroughly examined the transaction history and metadata connected to it. Next, they implemented a series of advanced recovery techniques, using cutting-edge software tools to bypass the access barriers that had left me locked out. The entire process was shrouded in secrecy and discretion, with the (TRUST GEEKS HACK EXPERT) team working tirelessly to protect the confidentiality of my sensitive financial information. After what felt like an eternity of nervous anticipation, the day finally arrived when I received the triumphant notification – my wallet had been successfully restored, and all of my precious digital assets had been returned to my control. The sense of relief was indescribable, as I could finally breathe easy knowing that the fruits of my financial discipline had been safeguarded. While the experience of losing access to my wallet was undoubtedly traumatic, (TRUST GEEKS HACK EXPERT) intervention allowed me to emerge from the ordeal with my cryptocurrency holdings intact, and a renewed appreciation for the importance of proactive digital asset management. You can contact Them through EMAIL: [email protected] - TELEGRAM: TRUSTGEEKSHACKEXPERT
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22.11.24 23:43 teresaborja

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24.11.24 02:21 [email protected]

I never could have imagined the nightmare of losing access to my digital wallet. All of my cryptocurrency holdings were abruptly imprisoned, inaccessible, and appeared to be lost forever following a catastrophic hardware breakdown. Years of meticulous investment and careful saving were reduced to nothing more than strings of code that I could no longer control, and I could feel the dread and sorrow that swept through me at that very instant. Thankfully, during my worst moment, I came into (TRUST GEEKS HACK EXPERT), a professional service devoted to recovering lost or inaccessible digital data. With optimism, I went out to their team of skilled technologists, laying bare the full nature of my issue. What followed was a laborious, multi-step process that required an almost surgical level of digital forensics and Bitcoin skill. In order to create a thorough profile of my wallet's contents and activities, the (TRUST GEEKS HACK EXPERT) team first thoroughly examined the transaction history and metadata connected to it. Next, they implemented a series of advanced recovery techniques, using cutting-edge software tools to bypass the access barriers that had left me locked out. The entire process was shrouded in secrecy and discretion, with the (TRUST GEEKS HACK EXPERT) team working tirelessly to protect the confidentiality of my sensitive financial information. After what felt like an eternity of nervous anticipation, the day finally arrived when I received the triumphant notification – my wallet had been successfully restored, and all of my precious digital assets had been returned to my control. The sense of relief was indescribable, as I could finally breathe easy knowing that the fruits of my financial discipline had been safeguarded. While the experience of losing access to my wallet was undoubtedly traumatic, (TRUST GEEKS HACK EXPERT) intervention allowed me to emerge from the ordeal with my cryptocurrency holdings intact, and a renewed appreciation for the importance of proactive digital asset management. You can contact Them through EMAIL: [email protected] - TELEGRAM: TRUSTGEEKSHACKEXPERT
25.11.24 02:19 briankennedy

COMMENT ON I NEED A HACKER TO RECOVER MONEY FROM BINARY TRADING. HIRE FASTFUND RECOVERY
25.11.24 02:20 briankennedy

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26.11.24 21:59 [email protected]

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26.11.24 23:12 rickrobinson8

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26.11.24 23:12 rickrobinson8

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27.11.24 00:39 [email protected]

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27.11.24 09:10 Michal Novotny

The biggest issue with cryptocurrency is that it is unregulated, wh ich is why different people can come up with different fake stories all the time, and it is unfortunate that platforms like Facebook and others only care about the money they make from them through ads. I saw an ad on Facebook for Cointiger and fell into the scam, losing over $30,000. I reported it to Facebook, but they did nothing until I discovered deftrecoup . c o m from a crypto community; they retrieved approximately 95% of the total amount I lost.
01.12.24 17:21 KollanderMurdasanu

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02.12.24 23:02 ytre89

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04.12.24 22:24 andreygagloev

When I first heard about Bitcoin back in 2018, I was skeptical. The idea of a decentralized, digital currency seemed too good to be true. But I was intrigued as I learned more about the technology behind it and its potential. I started small, investing just a few hundred dollars, dipping my toes into the cryptocurrency waters. At first, it was exhilarating to watch the value of my investment grow exponentially. I felt like I was part of the future, an early adopter of this revolutionary new asset. But that euphoria was short-lived. One day, I logged into my digital wallet only to find it empty - my Bitcoin had vanished without a trace. It turned out that the online exchange I had trusted had been hacked, and my funds were stolen. I was devastated, both financially and emotionally. All the potential I had seen in Bitcoin was tainted by the harsh reality that with decentralization came a lack of regulation and oversight. My hard-earned money was gone, lost to the ether of the digital world. This experience taught me a painful lesson about the price of trust in the uncharted territory of cryptocurrency. While the technology holds incredible promise, the risks can be catastrophic if you don't approach it with extreme caution. My Bitcoin investment gamble had failed, and I was left to pick up the pieces, wiser but poorer for having placed my faith in the wrong hands. My sincere appreciation goes to MUYERN TRUST HACKER. You are my hero in recovering my lost funds. Send a direct m a i l ( muyerntrusted ( @ ) mail-me ( . )c o m ) or message on whats app : + 1 ( 4-4-0 ) ( 3 -3 -5 ) ( 0-2-0-5 )
12.12.24 00:35 amandagregory

HOW TO HIRE A HACKER TO RECOVER STOLEN BITCOIN WITH FASTFUND RECOVERY
12.12.24 00:35 amandagregory

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27.12.24 00:21 swiftdream

I lost about $475,000.00 USD to a fake cryptocurrency trading platform a few weeks back after I got lured into the trading platform with the intent of earning a 15% profit daily trading on the platform. It was a hell of a time for me as I could hardly pay my bills and got me ruined financially. I had to confide in a close friend of mine who then introduced me to this crypto recovery team with the best recovery SWIFTDREAM i contacted them and they were able to completely recover my stolen digital assets with ease. Their service was superb, and my problems were solved in swift action, It only took them 48 hours to investigate and track down those scammers and my funds were returned to me. I strongly recommend this team to anyone going through a similar situation with their investment or fund theft to look up this team for the best appropriate solution to avoid losing huge funds to these scammers. Send complaint to Email: info [email protected]
31.12.24 04:53 Annette_Phillips

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04.01.25 04:56 florencestella

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04.01.25 04:57 florencestella

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06.01.25 19:09 michaeljordan15

We now live in a world where most business transactions are conducted through Bitcoin and cryptocurrency. With the rapid growth of digital currencies, everyone seems eager to get involved in Bitcoin and cryptocurrency investments. This surge in interest has unfortunately led to the rise of many fraudulent platforms designed to exploit unsuspecting individuals. People are often promised massive profits, only to lose huge sums of money when they realize the platform they invested in was a scam. contact with WhatsApp: +1 (443) 859 - 2886 Email @ digitaltechguard.com Telegram: digitaltechguardrecovery.com website link:: https://digitaltechguard.com This was exactly what happened to me five months ago. I was excited about the opportunity to invest in Bitcoin, hoping to earn a steady return of 20%. I found a platform that seemed legitimate and made my investment, eagerly anticipating the day when I would be able to withdraw my earnings. When the withdrawal day arrived, however, I encountered an issue. My bank account was not credited, despite seeing my balance and the supposed profits in my account on the platform. At first, I assumed it was just a technical glitch. I thought, "Maybe it’s a delay in the system, and everything will be sorted out soon." However, when I tried to contact customer support, the line was either disconnected or completely unresponsive. My doubts started to grow, but I wanted to give them the benefit of the doubt and waited throughout the day to see if the situation would resolve itself. But by the end of the day, I realized something was terribly wrong. I had been swindled, and my hard-earned money was gone. The realization hit me hard. I had fallen victim to one of the many fraudulent Bitcoin platforms that promise high returns and disappear once they have your money. I knew I had to act quickly to try and recover what I had lost. I started searching online for any possible solutions, reading reviews and recommendations from others who had faced similar situations. That’s when I came across many positive reviews about Digital Tech Guard Recovery. After reading about their success stories, I decided to reach out and use their services. I can honestly say that Digital Tech Guard Recovery exceeded all my expectations. Their team was professional, efficient, and transparent throughout the process. Within a short time, they helped me recover a significant portion of my lost funds, which I thought was impossible. I am incredibly grateful to Digital Tech Guard Recovery for their dedication and expertise in helping me get my money back. If you’ve been scammed like I was, don’t lose hope. There are solutions, and Digital Tech Guard Recovery is truly one of the best. Thank you, Digital Tech Guard Recovery! You guys are the best. Good luck to everyone trying to navigate this challenging space. Stay safe.
18.01.25 12:41 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance, contact [email protected] Capital Crypto Recover on Telegram OR Call Number +1 (336)390-6684 via email: [email protected]
18.01.25 12:41 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance, contact [email protected] Capital Crypto Recover on Telegram OR Call Number +1 (336)390-6684 via email: [email protected]
20.01.25 15:39 patricialovick86

How To Recover Your Bitcoin Without Falling Victim To Scams: A Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact: [email protected]
22.01.25 21:43 DoraJaimes23

Recovery expert. I lost my bitcoin to fake blockchain impostors on Facebook, they contacted me as blockchain official support and i fell stupidly for their mischievous act, this made them gain access into my blockchain wallet whereby 7.0938 btc was stolen from my wallet in total .I was almost in a comma and dumbfounded because this was all my savings i relied on . Then I made a research online and found a recovery expert , with the contact address- { RECOVERYHACKER101 (@) GMAIL . COM }... I wrote directly to the specialist explaining my loss. Hence, he helped me recover a significant part of my investment just after 2 days he helped me launch the recovery program , and the culprits were identified as well , all thanks to his expertise . I hope I have been able to help someone as well . Reach out to the recovery specialist to recover you lost funds from any form of online scam Thanks
23.01.25 02:36 [email protected]

After falling victim to a fraudulent Bitcoin mining scam, I found myself in a desperate situation. I had invested $50,000 into a cloud mining website called Miningpool, which turned out to be a complete scam. For months, I tried reaching out to the company, but I was unable to access my funds, and I quickly realized I had been taken for a ride. In my search for help, I came across TrustGeeks Hack Expert, a service that claimed to help people recover lost funds from crypto scams. Though skeptical at first, I decided to give them a try. Here’s my experience with their service.When I initially contacted TrustGeeks Hack Expert Email.. Trustgeekshackexpert{At}fastservice{Dot}com , I was understandably hesitant. Like many others, I had been tricked into believing my Bitcoin investments were legitimate, only to discover they were locked in a non-spendable wallet with no way of accessing them. However, after sharing my story and details about the scam, the team assured me they had handled similar cases and had the expertise to help. They requested basic information about my investment and began their investigation immediately. The recovery process was nothing short of professional. Unlike many other services that promise quick fixes but fail to deliver, TrustGeeks Hack Expert kept me informed at every stage. They regularly updated me on their progress and were completely transparent about the challenges they faced. There were moments when I wondered if the process would work, but the team’s professionalism and reassurance gave me hope. They were honest about the time it would take and did not make any unrealistic promises, which I truly appreciated. After several weeks of work, TrustGeeks Hack Expert successfully recovered not just my $50,000 investment, but also the so-called profits that had been locked away in the scam's non-spendable wallet. This was a huge relief, as I had resigned myself to the idea that I had lost everything. The entire recovery process was discreet and handled with the utmost care, ensuring that the scam company remained unaware of the recovery efforts, which helped prevent further complications. TeleGram iD. Trustgeekshackexpert & What's A p p +1 7 1 9 4 9 2 2 6 9 3
23.01.25 02:37 [email protected]

After falling victim to a fraudulent Bitcoin mining scam, I found myself in a desperate situation. I had invested $50,000 into a cloud mining website called Miningpool, which turned out to be a complete scam. For months, I tried reaching out to the company, but I was unable to access my funds, and I quickly realized I had been taken for a ride. In my search for help, I came across TrustGeeks Hack Expert, a service that claimed to help people recover lost funds from crypto scams. Though skeptical at first, I decided to give them a try. Here’s my experience with their service.When I initially contacted TrustGeeks Hack Expert Email.. Trustgeekshackexpert{At}fastservice{Dot}com , I was understandably hesitant. Like many others, I had been tricked into believing my Bitcoin investments were legitimate, only to discover they were locked in a non-spendable wallet with no way of accessing them. However, after sharing my story and details about the scam, the team assured me they had handled similar cases and had the expertise to help. They requested basic information about my investment and began their investigation immediately. The recovery process was nothing short of professional. Unlike many other services that promise quick fixes but fail to deliver, TrustGeeks Hack Expert kept me informed at every stage. They regularly updated me on their progress and were completely transparent about the challenges they faced. There were moments when I wondered if the process would work, but the team’s professionalism and reassurance gave me hope. They were honest about the time it would take and did not make any unrealistic promises, which I truly appreciated. After several weeks of work, TrustGeeks Hack Expert successfully recovered not just my $50,000 investment, but also the so-called profits that had been locked away in the scam's non-spendable wallet. This was a huge relief, as I had resigned myself to the idea that I had lost everything. The entire recovery process was discreet and handled with the utmost care, ensuring that the scam company remained unaware of the recovery efforts, which helped prevent further complications. TeleGram iD. Trustgeekshackexpert & What's A p p +1 7 1 9 4 9 2 2 6 9 3
23.01.25 14:20 nellymargaret

DUNAM CYBER SOLUTION BTC-ETH RECOVERY EXPERT
23.01.25 14:20 nellymargaret

DUNAM CYBER SOLUTION BTC-ETH RECOVERY EXPERT I had tried to secure my Bitcoin wallet, maybe a bit too aggressively, enabling every security feature imaginable: two-factor authentication, biometric verification, intricate passwords-the whole shebang. I wanted to make it impossible for anybody to get to my money. I tried to make this impregnable fortress of security and ended up locking myself out of my wallet with $700,000 in Bitcoin. It wasn't until I tried to access my wallet that I realized the trap I had set for myself. I was greeted with an endless series of security checks-passwords, codes, facial recognition, and more. I could remember parts of my multi-layered security setup but not enough to actually get in. In fact, my money was behind this digital fortress, and the more I tried to fix it, the worse it seemed to get. I kept tripping over my own layers of protection, unable to find a way back in. Panic quickly set in when I realized I had made it almost impossible for myself to access my own money. That is when I called DUNAMIS CYBER SOLUTION From that very first call, they reassured me that I wasn't the first person to make this kind of mistake and certainly wouldn't be the last. They listened attentively to my explanation and got to work straight away. Their team methodically began to untangle my overly complicated setup. Patience and expertise managed to crack each layer of security step by step until they had restored access to my wallet. [email protected] +13433030545 [email protected]
26.01.25 03:54 [email protected]

Losing access to my crypto wallet account was one of the most stressful experiences ever. After spending countless hours building up my portfolio, I suddenly found myself locked out of my account without access. To make matters worse, the email address I had linked to my wallet was no longer active. When I tried reaching out, I received an error message stating that the domain was no longer in use, leaving me in complete confusion and panic. It was as though everything I had worked so hard for was gone, and I had no idea how to get it back. The hardest part wasn’t just the loss of access it was the feeling of helplessness. Crypto transactions are often irreversible, and since my wallet held significant investments, the thought that my hard-earned money could be lost forever was incredibly disheartening. I spent hours scouring forums and searching for ways to recover my funds, but most of the advice seemed either too vague or too complicated to be of any real help. With no support from the wallet provider and my email account out of reach, I was left feeling like I had no way to fix the situation.That’s when I found out about Trust Geeks Hack Expert . I was hesitant at first, but after reading about their expertise in recovering lost crypto wallets, I decided to give them a try. I reached out to their team, and from the very beginning, they were professional, understanding, and empathetic to my situation. They quickly assured me that there was a way to recover my wallet, and they got to work immediately.Thanks to Trust Geeks Hack Expert , my wallet and funds were recovered, and I couldn’t be more grateful. The process wasn’t easy, but their team guided me through each step with precision and care. The sense of relief I felt when I regained access to my crypto wallet and saw my funds safely back in place was indescribable. If you find yourself in a similar situation, I highly recommend reaching out to Trust Geeks Hack Expert. contact Them through EMAIL: [email protected] + WEBSITE. HTTPS://TRUSTGEEKSHACKEXPERT.COM + TELE GRAM: TRUSTGEEKSHACKEXPERT
28.01.25 21:48 [email protected]

It’s unfortunate that many people have become victims of scams, and some are facing challenges accessing their Bitcoin wallets. However, there's excellent news! With Chris Wang, you can count on top-notch service that guarantees results in hacking. We have successfully helped both individuals and organizations recover lost files, passwords, funds, and more. If you need assistance, don’t hesitate—check out recoverypro247 on Google Mail! What specific methods does Chris Wang use to recover lost funds and passwords? Are there any guarantees regarding the success rate of the recovery services offered? What are the initial steps to begin the recovery process with recoverypro247? this things i tend to ask